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Theorem ndmovg 7145
Description: The value of an operation outside its domain. (Contributed by NM, 28-Mar-2008.)
Assertion
Ref Expression
ndmovg ((dom 𝐹 = (𝑅 × 𝑆) ∧ ¬ (𝐴𝑅𝐵𝑆)) → (𝐴𝐹𝐵) = ∅)

Proof of Theorem ndmovg
StepHypRef Expression
1 df-ov 6977 . 2 (𝐴𝐹𝐵) = (𝐹‘⟨𝐴, 𝐵⟩)
2 eleq2 2847 . . . . . 6 (dom 𝐹 = (𝑅 × 𝑆) → (⟨𝐴, 𝐵⟩ ∈ dom 𝐹 ↔ ⟨𝐴, 𝐵⟩ ∈ (𝑅 × 𝑆)))
3 opelxp 5439 . . . . . 6 (⟨𝐴, 𝐵⟩ ∈ (𝑅 × 𝑆) ↔ (𝐴𝑅𝐵𝑆))
42, 3syl6bb 279 . . . . 5 (dom 𝐹 = (𝑅 × 𝑆) → (⟨𝐴, 𝐵⟩ ∈ dom 𝐹 ↔ (𝐴𝑅𝐵𝑆)))
54notbid 310 . . . 4 (dom 𝐹 = (𝑅 × 𝑆) → (¬ ⟨𝐴, 𝐵⟩ ∈ dom 𝐹 ↔ ¬ (𝐴𝑅𝐵𝑆)))
6 ndmfv 6526 . . . 4 (¬ ⟨𝐴, 𝐵⟩ ∈ dom 𝐹 → (𝐹‘⟨𝐴, 𝐵⟩) = ∅)
75, 6syl6bir 246 . . 3 (dom 𝐹 = (𝑅 × 𝑆) → (¬ (𝐴𝑅𝐵𝑆) → (𝐹‘⟨𝐴, 𝐵⟩) = ∅))
87imp 398 . 2 ((dom 𝐹 = (𝑅 × 𝑆) ∧ ¬ (𝐴𝑅𝐵𝑆)) → (𝐹‘⟨𝐴, 𝐵⟩) = ∅)
91, 8syl5eq 2819 1 ((dom 𝐹 = (𝑅 × 𝑆) ∧ ¬ (𝐴𝑅𝐵𝑆)) → (𝐴𝐹𝐵) = ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 387   = wceq 1508  wcel 2051  c0 4172  cop 4441   × cxp 5401  dom cdm 5403  cfv 6185  (class class class)co 6974
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-ext 2743  ax-sep 5056  ax-nul 5063  ax-pow 5115  ax-pr 5182
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-3an 1071  df-tru 1511  df-ex 1744  df-nf 1748  df-sb 2017  df-mo 2548  df-eu 2585  df-clab 2752  df-cleq 2764  df-clel 2839  df-nfc 2911  df-ral 3086  df-rex 3087  df-rab 3090  df-v 3410  df-dif 3825  df-un 3827  df-in 3829  df-ss 3836  df-nul 4173  df-if 4345  df-sn 4436  df-pr 4438  df-op 4442  df-uni 4709  df-br 4926  df-opab 4988  df-xp 5409  df-dm 5413  df-iota 6149  df-fv 6193  df-ov 6977
This theorem is referenced by:  ndmov  7146  curry1val  7606  curry2val  7610  1div0  11098  repsundef  13988  cshnz  14012  cshnzOLD  14013  mamufacex  20717  mavmulsolcl  20879  mavmul0g  20881  iscau2  23598  1div0apr  28039  rrxsphere  44137
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